Bounded weak solutions to matrix drift-diffusion model for spin-coherent electron transport in semiconductors

The global-in-time existence and uniqueness of bounded weak solutions to a spinorial matrix drift-diffusion model for semiconductors is proved. Developing the electron density matrix in the Pauli basis, the coefficients (charge density and spin-vector density) satisfy a parabolic $4\times 4$ cross-diffusion system. The key idea of the existence proof is to work with different variables: the spin-up and spin-down densities as well as the parallel and perpendicular components of the spin-vector density with respect to the magnetization. In these variables, the diffusion matrix becomes diagonal. The proofs of the $L^\infty$ estimates are based on Stampacchia truncation as well as Moser- and Alikakos-type iteration arguments. The monotonicity of the entropy (or free energy) is also proved. Numerical experiments in one space dimension using a finite-volume discretization indicate that the entropy decays exponentially fast to the equilibrium state.